1、Chapter Ray-tracing and Gaussian theory of lenses 光线追迹及透镜的高斯理论 2.1 Ray tracing procedure 光线追迹过程 2.2 Magnification and Lagrange theorem 放大倍率和拉格朗日理论 2.3 Gaussian theory of lenses 透镜的高斯理论 2.4 Conjugate distance relationships 共轭距关系 1 Newtons equation 2 Gaussian equation 3 Nodal points 4 Longitudinal mag
2、nification PDF 文件使用 “pdfFactory Pro“ 试用版本创建 Purpose: determine the position, size of image (illumination) Problem: 1 How many cardinal points are there (principal, focal, nodal) What are their characters? 2 What is the relationship between the principal sections H, H (=+1) What is the relationship b
3、etween the focal length f, f ( nnff = ) 3 write down the equation about the f and fl 1kuhf = kkf uhl = (Back focal length) (Back focal distance) w yf = 4 how to express the Lagrange equation For a distant object () tgwftgfy = w For a lens in the air ( ww = , nn ) PDF 文件使用 “pdfFactory Pro“ 试用版本创建 2.1
4、 Ray tracing procedure 1 The type of Rays Rays in general fall into three classes: Meridional, paraxial, skew u Meridional ray lie in meridional plane ,which is the plane containing the lens axis and an object point lying to one side of the axis u Paraxial ray extremely close to the optical axis. Th
5、e image formed by paraxial ray is deemed to be a perfect image. u Skew rays do not lie in the meridian plane, but they pass in front of it or behind it and pierce the meridional plane at the spot diagram. 2 ray tracing We use a set of trigonometric formula a Z b C D PDF 文件使用 “pdfFactory Pro“ 试用版本创建
6、Meridional ray tracing The rules of sign 1) aperture angle ，u u If clockwise rotation takes us from axis to ray positive counterclockwise rotation negative 2) incidence angle and refracted angle II， Turn the ray to normal If clockwise rotation positive counterclockwise negative 3) ,r ll : from surfa
7、ce vertex to object A , image A It is reckoned: positive: if along light travel negative: if against light travel , yyhh z : From axis to object B , image B Positive: above the surface axis Negative: under the surface axis consider triangle ,according to sine law, we get ruLrI )sin(sin = So Ur rLI s
8、insin = (1) Apply the law of refraction to determine I InnI sinsin = (2) From UIUI +=+=f IUIU += (3) Consider triangle, similar to (1) PDF 文件使用 “pdfFactory Pro“ 试用版本创建 rLIrU= sinsin sin sin UIrrL += (4) If we known L, U, we can calculate L, U For plane surface , )sin(sin UI = ,R is infinite. )()(ULU
9、LhIUUI= IILUULL = Conclusion for different aperture angle U, the image distance L is different ,so a perfect spherical surface system has aberrations 3 paraxial rays Paraxial ray is in theory infinitely close to the optical axis, so uUiI=sinsin uirrliuiuinniur rli+=+=Surface transferring equation 11
10、12112udhhdll= 4 Abbe s invariable From inni = )()( urhnurhn = -u I I h n n -u PDF 文件使用 “pdfFactory Pro“ 试用版本创建 fhr nnhnuun = (1) rnnhlhnlhn = fnrnnlnln = (2) Abbe: Qlrnlrn = )11()11( (3) Conclusion: All the paraxial angles II， , ，u u have disappeared . This relation ( r nnlnln = ) shows that .All pa
11、raxial rays emerging from a given object point pass through the same image point. No matter how large is the angle u? 5 concave or convex mirror r 125., 177.6, 254., 41.8, d -8.9, -100, 134.2, 15.4, 1 n -1, -1.545, -1, 1, 1.545 The rule is: to list the surface in succession in the order the light st
12、rikes them, with correct axial separation d and index n Note that if the light is traveling from right to left: Both d and n must be entered with a negative sign. PDF 文件使用 “pdfFactory Pro“ 试用版本创建 2.2 magnification and Lagrange theorem Transverse magnification = (transverse dimension of the image y)/
13、 (transverse dimension of the object y) or (height of image y)/ (height of object y) Trace a paraxial ray from B, the top of the object, to the surface vertex and onto the top of the image at B lyI = lyI = From the law of refraction nInI = lynlyn = Multiply by h in both sides lhynlhny = Juynnyu = is
14、 known as the theorem of Lagrange = 222222111111 yunyunyunyun For complete system (nuy) product is optical invariable, so J is called the Lagrange Invariable. 11kkkunnuyy =b (1) If lens in the air 1kuu=b Because of 111111 , lhulhu = For single surface, the magnification is lnnlyy11 =b (2) PDF 文件使用 “
15、pdfFactory Pro“ 试用版本创建 For complete system with K surfaces kkkkkkkkkkllllllllnnlnlnlnlnlnlnlnlnyyyyyyyyyy32132113333222211113322111=b(3) PDF 文件使用 “pdfFactory Pro“ 试用版本创建 2.3 Gaussian theory of lenses 1 The four cardinal points Gauss met the problem of focal length for thick lenses by postulating fou
16、r cardinal points in any lens: two focal points F, F and two principal points H, H 1. A set of rays A.B.C. enter the left-hand end of the lens, and are parallel to the lens axis. 2. After pass through the lens, these rays cross the axis at various points. 3. By extending each ray backward or forward
17、 until the entering and emerging portion intersect, we can locate an equivalent refracting point for each ray, Q, R, P. The locus of all such points is equivalent refracting locus of lens. the locus within the paraxial region is a plane perpendicular to the lens axis called the principal plane H The
18、 image point for paraxial rays lies at the focal point F The axial distance from Hto F is called the focal length f Similarly we can determine another pair cardinal point H F, for light entering the lens parallel to the axis from the right and emerging to left. 2 Relation between principal planes tw
19、o principal planes are images of each other at unit magnification According to reversibility of the optical path We reverse the arrows on the ray B. We end up with two rays entering through R and emerging through Q. 3 Relation between the focal length PDF 文件使用 “pdfFactory Pro“ 试用版本创建 Suppose we plac
20、e a small object at front focal point F, and draw two rays from the top of the object into the lens fh=w fh=w ff=ww (1) Move the small object along the axis to H, the image is at H. Applying Lagrange equation to this pair of conjugates, we get nnhnhn = wwww (2) Combing the two equations, gives nnff
21、= (3) Focal length 1kuhf = Back focal distance kkf uhl = (4) 4 the Lagrange equation for a distant object yunnuy = has no meaning for an infinitely distant object ,for u=0,y= ,and the product of zero times infinity is indeterminate. We interpret Lagrange equation for a distant object. PDF 文件使用 “pdfF
22、actory Pro“ 试用版本创建 hnylhnyunhnylhnnuy ww= ww nn = (1) wwwtgfnntguhnnyhntgyun=(2) For a lens in the air ),( ww = nn ww tgftgfy = Two expressions for focal length: wyfuhf=is used for paraxial ray 5 lens power The power of a single refracting surface is defined as rnn /)( =f The power of a complete len
23、s system is defined as fn=f n : the refractive index in the image space nnff = similar fn=f The usual unit of power is diopter For a lens with surface number K kkkkkk hununhununhununfff=222222111111 Add all these expressions together, the sum becomes merely = kkk hunun111 f For the case of an infini
24、tely instant object = kkk hhhunu11111,0 f PDF 文件使用 “pdfFactory Pro“ 试用版本创建 = k hh111 ff 2.4 Conjugate distance relationships Purpose: determine the position and size of the image of a given object 1 distance from focal points F, F By similar triangles fyfHQxy=xfyy = b (1) ABF PHF fyfHPxy = fxyy =b (
25、2) Newtons equation fxxf = 2nnfxxffxx=)( nnff = PDF 文件使用 “pdfFactory Pro“ 试用版本创建 2 distance from principal points H,H 1)(,=+=+=lflffffffllfllffflflflxflxImage magnification is lnnlyy =b In the air or homogenous medium llfll111=b3 nodal points Nodal points are a pair of conjugate points having the pr
26、operty that angular magnification=1 (Paraxial ray entering toward the first surface will emerge from second at the same slope) nnyyynyn=bwwwwNewtons formula PDF 文件使用 “pdfFactory Pro“ 试用版本创建 )(nnfnnxnnfffxnnfxyy=bSimilarly )(fxfnnnnnnfxnnxf=b4 longitudinal magnification L, M is the ratio of the axial
27、 dimension of an image to the corresponding axial dimension of the object dxdx=a or xx= a , xdx may be the physical sizes of image and object ,maybe a movement of object and image along the axis . Differentiating formula rnnlnln = 22222baannllnndldldllndlln=Multiplying numerator and denominator by h
28、2, gives 2222nUdlndlUUnnUdldl=a Magnification invariant For large axial dimension PDF 文件使用 “pdfFactory Pro“ 试用版本创建 BBAAfxfxbb= When object moves from A to B The change in image distance ABis given by )()( ABBA fxxBA bb = Similarly the change in the object distance AB is BABAABBABABBAAffABBAfxxABfxfxbbbbbbabbbb=)11()()11()(,Application optical bench PDF 文件使用 “pdfFactory Pro“ 试用版本创建
